Derivative of cross-product of two vectors \frac{d}{dt} [\vec{u(t)} \times \vec{v(t)}],

Gregory Jones

Gregory Jones

Answered question

2021-12-16

Derivative of cross-product of two vectors ddt[u(t)×v(t)], is it possible to find the cross-product of the two vectors first before differentiating?

Answer & Explanation

xandir307dc

xandir307dc

Beginner2021-12-17Added 35 answers

This expression can be assessed in one of two ways: 
Finding the cross-product first, then differentiating it, is an option.
Alternately, you could employ the product rule, which is compatible with the cross product:
d dt (u×v)=du dt ×v+u×dv dt  
The method to use will depend on the issue at hand. For instance, Frenet Serret formulas are derived using the product rule.

Esta Hurtado

Esta Hurtado

Beginner2021-12-18Added 39 answers

Starting with the fundamentals:
u(t+δt)×v(t+δt)u(t)×v(t)=u(t+δt)×v(t+δt)u(t)×v(t+δt)+
=u(t)×v(t+δt)u(t)×v(t)=
=[u(t+δt)u(t)]×v(t+δt)+
=u(t)×[v(t+δt)v(t)]
Now divide by δ and take limit as δt0
As opposed to that,
ddt[ijkvxvyvzuxuyuz]=[ijkdvxdtdvydtdvzdtuxuyuz]+[ijkvxvyvzduzdtduydtduzdt]
Use a determinant's differentiation rule. Its use in establishing Abel's identification is one of its many applications.

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